Analytical Chem istry - DePauw University

Chapter 4 Evaluating Analytical Data97Table 4.15 Values of t for a 95% Confidence IntervalDegrees ofDegrees ofFreedomtFreedomt1 12.706 12 2.1792 4.303 14 2.1453 3.181 16 2.1204 2.776 18 2.1015 2.571 20 2.0866 2.447 30 2.0427 2.365 40 2.0218 2.306 60 2.0009 2.262 100 1.98410 2.228 ∞ 1.960Example 4.15What are the 95% confidence intervals for the two samples of pennies inTable 4.11?So l u t i o nThe mean and standard deviation for first experiment are, respectively,3.117 g and 0.051 g. Because the sample consists of seven measurements,there are six degrees of freedom. The value of t from Table 4.15, is 2.447.Substituting into equation 4.12 gives2. 447×0.051 gµ= 3.117 g ±= 3. 117 g±0.047 g7For the second experiment the mean and standard deviation are 3.081 gand 0.073 g, respectively, with four degrees of freedom. The 95% confidenceinterval is2. 776×0.037 gµ= 3.081 g ±= 3. 081 g±0.046 g5Based on the first experiment, there is a 95% probability that the population’smean is between 3.070 to 3.164 g. For the second experiment,the 95% confidence interval spans 3.035 g–3.127 g. The two confidenceintervals are not identical, but the mean for each experiment is containedwithin the other experiment’s confidence interval. There also is an appreciableoverlap of the two confidence intervals. Both of these observationsare consistent with samples drawn from the same population.Our comparison of these two confidenceintervals is rather vague and unsatisfying.We will return to this point in the nextsection, when we consider a statistical approachto comparing the results of experiments.